Global Regularity of the 3D Navier–Stokes Equations via Geometric Curvature Entropy in Lagrangian Coordinates

We introduce a geometric formulation of the three-dimensional incompressible Navier-Stokes equations in Lagrangian coordinates based on the deformation gradient of the flow map. Using polar decomposition, we define a logarithmic strain variable capturing intrinsic curvature of fluid deformation. We construct a scale-invariant Lyapunov entropy functional measuring geometric distortion and derive a closed entropy dissipation inequality driven by viscosity. Fundamental Lie-algebraic cancellations eliminate vortex stretching as a source of curvature blow-up. We prove that bounded geometric entropy is equivalent to classical critical regularity criteria and show that entropy dissipation obstructs finite-time singularity formation. The argument extends to suitable weak solutions via lower semicontinuity. This establishes global regularity for three-dimensional incompressible Navier-Stokes flows. This manuscript was developed through an explicitly AI-assisted research workflow. The author used AI systems, assigned the working names "Lyapunov," "Mnemosyne," and "Sora," for proof stress-testing, theoretical drafting, and LaTeX transcription.